Paper · May 21, 2020

Self-similar solutions to the mean curvature flow in

Benedito Leandro, Rafael Novais, Hiuri F. S. dos Reis

math.DGWider flowsarXiv:2005.10688v4
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As on arXiv

Abstract

In this paper we make an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in . We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolutions under a homothetic helicoidal motion in in terms of the curvature of the generating curve. Finally, we characterize the self-similar solutions for the MCF by cylindrical surfaces under a homothetic helicoidal motion in . Explicit families of exact solutions for the MCF by cylindrical surfaces in are also given.

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Submitted
May 21, 2020
Last revised
Oct 6, 2021 (v4)
Subjects
math.DG
Journal
Differential Geometry and its Applications 2023
DOI
10.1016/j.difgeo.2023.101985
Mean curvature flowRotational symmetry & warped products

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Some results on translating solitons of the mean curvature flow (Jesús Pérez-García, 2016)4Pérez-García 2016 A family of MCF solutions for the Heisenberg Group (Benedito Leandro, Adriana Araujo Cintra and Hiuri Fellipe dos Santos Reis, 2019)2Leandro 2019 Another proof of the existence of homothetic solitons of the inverse mean curvature flow (Shu-Yu Hsu, 2020)8Hsu 2020 Ruled surfaces of generalized self-similar solutions of the mean curvature flow (Rafael López, 2020)3López 2020 Translators of the Gauss curvature flow (Muhittin Evren Aydin and Rafael López, 2022)5Aydin 2022 Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces (Hiuri dos Reis, Benedito Leandro and Rafael Novais, 2022)1dos Reis 2022 Gauss Curvature Flow on Surfaces of Revolution – The Noncompact Case (Thalia D. Jeffres and Leonardo Solanilla, 2024)7Jeffres 2024 Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space (Ronaldo F. de Lima and Álvaro K. Ramos, 2026)6de Lima 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces

    Hiuri dos Reis, Benedito Leandro and Rafael NovaisWider flows

    Same authors Benedito Leandro, Rafael Novais, Hiuri F. S. dos Reis · Shared terms “revolution”

  2. A family of MCF solutions for the Heisenberg Group

    Benedito Leandro, Adriana Araujo Cintra and Hiuri Fellipe dos Santos ReisWider flows

    Topics Mean curvature flow · Same author Benedito Leandro · Shared terms “MCF”, “ruled surfaces”

  3. Ruled surfaces of generalized self-similar solutions of the mean curvature flow

    Rafael LópezWider flows

    Topics Mean curvature flow · Shared terms “ruled surfaces”, “cylindrical surfaces”

  4. Some results on translating solitons of the mean curvature flow

    Jesús Pérez-GarcíaWider flows

    Topics Rotational symmetry & warped products · Shared terms “revolution”

  5. Translators of the Gauss curvature flow

    Muhittin Evren Aydin and Rafael LópezWider flows

    Topics Rotational symmetry & warped products · Shared terms “helicoidal motions”

  6. Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

    Ronaldo F. de Lima and Álvaro K. RamosWider flows

    Topics Rotational symmetry & warped products · Shared terms “MCF”

  7. Gauss Curvature Flow on Surfaces of Revolution – The Noncompact Case

    Thalia D. Jeffres and Leonardo SolanillaWider flows

    Topics Rotational symmetry & warped products · Shared terms “revolution”

  8. Another proof of the existence of homothetic solitons of the inverse mean curvature flow

    Shu-Yu HsuWider flows

    Topics Rotational symmetry & warped products · Shared terms “homothetic”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.